已知函数f(x)=4sinxsin^2(∏/4+x/2)+cos2x,x∈[π/6,4π/3],求f(x)最大值与最小值,并求对应x的值

题目

已知函数f(x)=4sinxsin^2(∏/4+x/2)+cos2x,x∈[π/6,4π/3],求f(x)最大值与最小值,并求对应x的值

答案

∵f(x)=4sinxsin²(π/4+x/2)+cos(2x)
=2sinx[2sin²(π/4+x/2)]+cos(2x)
=2sinx[1-cos(2(π/4+x/2))]+cos(2x)
=2sinx[1-cos(π/2+x)]+cos(2x)
=2sinx(1+sinx)+1-2sin²x
=2sinx+1
∴令f '(x)=2cosx=0,则 x=π/2 (∵x∈[π/6,4π/3])
∵f(π/6)=2sin(π/6)+1=2
f(π/2)=2sin(π/2)+1=3
f(4π/3)=2sin(4π/3)+1=1-√3
∴f(x)最大值是f(π/2)=3,最小值是f(4π/3)=1-√3.